Exercises: Prove theorems about triangles

A

Warm-Up: Review What You Know

These problems review skills you already have.

1.

Lines $\ell$ and $m$ are parallel, cut by a transversal $t$ . Which theorem guarantees that the alternate interior angles are congruent?

A.

Vertical Angles Theorem

B.

Alternate Interior Angles Theorem

C.

Corresponding Angles Postulate

D.

Same-Side Interior Angles Theorem

2.

Triangle $ABC$ has $AB = AC$ . An angle bisector is drawn from $A$ to point $D$ on $\overline{BC}$ . Which congruence criterion proves $\triangle ABD \cong \triangle ACD$ ?

A.

SSS (Side-Side-Side)

B.

ASA (Angle-Side-Angle)

C.

SAS (Side-Angle-Side)

D.

AAS (Angle-Angle-Side)

3.

Point $D$ is the midpoint of $\overline{AB}$ , where $A = (0, 0)$ and $B = (8, 0)$ . What are the coordinates of $D$ ?

A.

$(4, 0)$

B.

$(2, 0)$

C.

$(8, 0)$

D.

$(0, 4)$

B

Fluency Practice

Apply the triangle theorems directly to find missing values.

1.

In triangle $PQR$ , $m\angle P = 47^\circ$ and $m\angle Q = 68^\circ$ . Find $m\angle R$ .

2.

Which theorem from HSG.CO.C.9 is the essential step in the proof of the Triangle Angle Sum Theorem?

A.

Vertical Angles Theorem — vertical angles at a point sum to 180°

B.

Alternate Interior Angles Theorem — when a transversal cuts parallel lines, alternate interior angles are congruent

C.

Exterior Angle Theorem — an exterior angle equals the sum of the two remote interior angles

D.

Corresponding Angles Postulate — corresponding angles formed by a transversal are supplementary

3.

Isosceles triangle $ABC$ has $AB = AC$ . The vertex angle measures $\angle A = 50^\circ$ . Find $m\angle B$ .

4.

In equilateral triangle $XYZ$ , all three sides are equal. What is $m\angle X$ ?

5.

In triangle $ABC$ , $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{AC}$ . If $BC = 24$ , find the length of midsegment $\overline{DE}$ .

C

Mixed Practice

These problems apply the triangle theorems in varied formats.

D

Word Problems

Read each problem carefully and apply the appropriate triangle theorem.

1.

A triangular brace is used in a construction project. Two angles of the triangle measure $38^\circ$ and $74^\circ$ .

What is the measure of the third angle of the brace?

2.

A surveyor marks triangle $ABC$ on a map with coordinates $A(0, 0)$ , $B(10, 0)$ , $C(4, 8)$ . The surveyor wants to place markers at the midpoints of two sides.

1.

The midpoint $D$ of $\overline{AB}$ is at $(5, 0)$ and the midpoint $E$ of $\overline{AC}$ is at $(2, 4)$ . Find the length of $\overline{BC}$ using the distance formula. Round to the nearest tenth.

2.

Using the Triangle Midsegment Theorem, what should the length of $\overline{DE}$ be? Verify by computing $DE$ directly using the distance formula.

3.

Triangle $KLM$ has vertices $K(0, 0)$ , $L(9, 0)$ , $M(3, 6)$ . The centroid $G$ lies on the median from $K$ to the midpoint of $\overline{LM}$ .

Find the coordinates of the centroid $G$ of triangle $KLM$ .

E

Find the Mistake

Each problem shows a student's work that contains an error. Identify and explain the mistake.

1.

Jordan is working on a geometry problem about a quadrilateral $ABCD$ with angles $\angle A = 80^\circ$ , $\angle B = 95^\circ$ , and $\angle C = 70^\circ$ . Jordan writes:

"Since the angles of any polygon sum to $180^\circ$ , the missing angle is $\angle D = 180^\circ - 80^\circ - 95^\circ - 70^\circ = -65^\circ$ ."

What error did Jordan make?

A.

Jordan applied the Triangle Angle Sum Theorem (180°) to a quadrilateral. Quadrilaterals have an angle sum of 360°.

B.

Jordan subtracted incorrectly — the arithmetic is wrong.

C.

Jordan forgot to include angle $\angle D$ in the equation.

D.

No line parallel to a side was drawn, so the proof is incomplete.

2.

Priya is writing a proof that the base angles of isosceles triangle $ABC$ are congruent, given $AB = AC$ . Her proof reads:

"Statement: $AB = AC$ . Reason: Given.
Statement: $\triangle ABC \cong \triangle ABC$ . Reason: Reflexive property.
Statement: $\angle B \cong \angle C$ . Reason: CPCTC."

What is wrong with Priya's proof?

A.

Priya used the wrong congruence criterion — she should use SSS, not CPCTC.

B.

Priya cannot use CPCTC without first establishing that two different triangles are congruent. Her proof does not create a second triangle to compare.

C.

The Reflexive Property does not apply in triangle proofs.

D.

Priya should have started by drawing the angle bisector of $\angle B$ , not $\angle A$ .

F

Challenge Problems

These problems require multi-step reasoning. Show all your work.

1.

An exterior angle of a triangle is formed by extending one side. Prove that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. Use the Triangle Angle Sum Theorem in your proof.